Non-structure in lambda^{++} using instances of WGCH

TL;DR

This paper advances the study of non-structure phenomena in abstract elementary classes by refining techniques using weak diamond principles in lambda^+ and lambda^{++}, aiming for clearer examples and improved results.

Contribution

It improves and extends previous non-structure results in a.e.c. by employing instances of WGCH, making the context closer to examples and enhancing transparency.

Findings

Enhanced non-structure results in lambda^+ and lambda^{++}

Better alignment of theoretical context with concrete examples

Progress in understanding the positive theory of low frameworks

Abstract

We try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in lambda^+ and lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. Comparing with math.LO/9805146 we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of "low frameworks" like almost good lambda-frames.